Laplacian in planar parabolic coordinates (${\Delta }_{xy}f=(g_{uu}+g_{vv})/(u^2+v^2)$)

For planar parabolic coordinates and $g(u,v)=f(x(u,v),y(u,v))$, the multivariable chain rule gives $g_u=u{f}_x+v{f}_y$ and $g_v=-v{f}_x+u{f}_y$. Differentiating once more and adding cancels the first partial derivatives and the mixed partial derivatives:Consequently, on every local coordinate patch away from the origin, $f$ is a harmonic function if and only if $g$ is a harmonic function. The Laplacian changes by a positive scale factor, so solutions of the Laplace equation are preserved. The restriction away from the origin is needed to divide by the Jacobian determinant.